Weierstrass factorization theorem

Weierstrass factorization theorem

In mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes. In addition, every sequence tending to infinity has an associated entire function with zeroes at precisely the points of that sequence.

A second form extended to meromorphic functions allows one to consider a given meromorphic function as a product of three factors: the function's poles, zeroes, and an associated non-zero holomorphic function.

Motivation

The consequences of the fundamental theorem of algebra are twofold.citation|last=Knopp|first=K.|contribution=Weierstrass's Factor-Theorem|title=Theory of Functions, Part II|location=New York|publisher=Dover|pages=1–7|year=1996.] Firstly, any finite sequence,{c_n}, in the complex plane has an associated polynomial that has zeroes precisely at the points of that sequence:

:,prod_n (z-c_n).

Secondly, any polynomial function in the complex plane, p(z), has a factorization

:,p(z)=aprod_n(z-c_n),

where "a" is a non-zero constant and "c""n" are the zeroes of "p".

The two forms of the Weierstrass factorization theorem can be thought of as extensions of the above to entire functions. The necessity of extra machinery is demonstrated when one considers whether the product

:,prod_n (z-c_n)

defines an entire function if the sequence, {c_n}, is not finite. The answer is never, because the now-infinite product will not converge. Thus one cannot, in general, define an entire function from a sequence of prescribed zeroes or represent an entire function by its zeroes using the expressions yielded by the fundamental theorem of algebra.

A necessary condition for convergence of the infinite product in question is: each factor, (z-c_n) , must approach 1 as n oinfty. So it stands to reason that one should seek a function that could be 0 at a prescribed point, yet remain near 1 when not at that point and furthermore introduce no more zeroes than those prescribed. Enter the genius of Weierstrass' "elementary factors". These factors serve the same purpose as the factors, (z-c_n) , above.

The elementary factors

These are also referred to as "primary factors".citation|last=Boas|first=R. P.|title=Entire Functions|publisher=Academic Press Inc.|location=New York|year=1954|isbn=0821845055|oclc=6487790, chapter 2.]

For n in mathbb{N}, define the "elementary factors":citation|last=Rudin|first=W.|title=Real and Complex Analysis|edition=3rd|publisher=McGraw Hill|location=Boston|pages=301–304|year=1987|isbn=0070542341|oclc=13093736.]

:E_n(z) = egin{cases} (1 -z) & mbox{if }n=0, \ (1-z)exp left( frac{z^1}{1}+frac{z^2}{2}+cdots+frac{z^n}{n} ight) & mbox{otherwise}. end{cases}

Their utility lies in the following lemma:

Lemma (15.8, Rudin) for |"z"| ≤ 1, "n" ∈ No

:vert 1 - E_n(z) vert leq vert z vert^{n+1}.

The two forms of the theorem

equences define holomorphic functions

Sometimes called the Weierstrass theoremMathWorld | urlname=WeierstrasssTheorem | title=Weierstrass's Theorem]

If lbrace z_i brace_i subset mathbb{C}-{0} is a sequence such that:
#vert z_i vert ightarrow infty as i ightarrow infty
#there is a sequence, lbrace p_i brace_i subset mathbb{N}_o, such that for all "r" > 0, sum_{i} left( frac{r}{vert z_i vert} ight)^{1+p_i} < infty.Then there exists an entire function that has (only) zeroes at every point of lbrace z_i brace; in particular, "P" is such a function:

: P(z)=prod_{i=1}^infty E_{p_i}left(frac{z}{z_i} ight).

*The theorem generalizes to: sequences in open subsets (and hence regions) of the Riemann sphere have associated functions that are holomorphic in those subsets and have zeroes at the points of the sequence.
*Note also that the case given by the fundamental theorem of algebra is incorporated here. If the sequence, { z_i } is finite then setting p_i = 0 suffices for convergence in condition 2, and we obtain: , P(z) = prod_n (z-z_n).

Holomorphic functions can be factored

Sometimes called the Weierstrass Product/Factor/Factorization theorem.MathWorld | urlname=WeierstrassProductTheorem | title=Weierstrass Product Theorem] Sometimes called the Hadamard Factorization theorem; for example c.f. Boas.

If "f" is a function holomorphic in a region, Omega, with zeroes at every point of lbrace z_i brace_i subset mathbb{C}-{0}then there exists an entire function "g", and a sequence lbrace p_i brace_i subset mathbb{R}_o^+ such that:


f(z)=e^{g(z)} prod_{i=1}^infty E_{p_i}left(frac{z}{z_i} ight)

References

See also

* Entire function


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Weierstrass preparation theorem — In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to multiplication by a function not zero at P, a polynomial… …   Wikipedia

  • Weierstrass theorem — Several theorems are named after Karl Weierstrass. These include: *The Weierstrass approximation theorem, also known as the Stone Weierstrauss theorem *The Bolzano Weierstrass theorem, which ensures compactness of closed and bounded sets in R n… …   Wikipedia

  • Karl Weierstrass — Infobox Scientist name = Karl Weierstrass |300px caption = Karl Theodor Wilhelm Weierstrass (Weierstraß) birth date = birth date|1815|10|31|mf=y birth place = Ostenfelde, Westphalia death date = death date and age|1897|2|19|1815|10|31|mf=y death… …   Wikipedia

  • Mittag-Leffler's theorem — In complex analysis, Mittag Leffler s theorem concerns the existence of meromorphic functions with prescribed poles. It is sister to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. It …   Wikipedia

  • List of mathematics articles (W) — NOTOC Wad Wadge hierarchy Wagstaff prime Wald test Wald Wolfowitz runs test Wald s equation Waldhausen category Wall Sun Sun prime Wallenius noncentral hypergeometric distribution Wallis product Wallman compactification Wallpaper group Walrasian… …   Wikipedia

  • Gamma function — For the gamma function of ordinals, see Veblen function. The gamma function along part of the real axis In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its… …   Wikipedia

  • List of complex analysis topics — Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied …   Wikipedia

  • Entire function — In complex analysis, an entire function, also called an integral function, is a complex valued function that is holomorphic over the whole complex plane. Typical examples of entire functions are the polynomials and the exponential function, and… …   Wikipedia

  • Zero (complex analysis) — In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. Contents 1 Multiplicity of a zero 2 Existence of zeros 3 Properties …   Wikipedia

  • Polygamma function — In mathematics, the polygamma function of order m is defined as the ( m + 1)th derivative of the logarithm of the gamma function::psi^{(m)}(z) = left(frac{d}{dz} ight)^m psi(z) = left(frac{d}{dz} ight)^{m+1} lnGamma(z).Here :psi(z) =psi^{(0)}(z) …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”